Example 1 of 7 · Area of a square

AI HL · Mathematics

A square floor tile has side length 50 cm, correct to the nearest 10 cm. Find the maximum percentage error in the area of the tile.

Smallest true value

2025 45 45
\[\begin{aligned}T_2&=45^2=2025\\[6pt]\class{mjx-pe-s6}{\%\text{ error}}&\class{mjx-pe-s6}{{}=\left|\frac{2500-2025}{2025}\right|\times100}\\[6pt]&\class{mjx-pe-s6}{{}=\class{mjx-pe-key}{\boldsymbol{23.5\%}}}\end{aligned}\]

Approximate value

2500 50 50
\[\begin{aligned}A&=50^2=2500\\[6pt]\class{mjx-pe-s5}{\%\text{ error}}&\class{mjx-pe-s5}{{}=\left|\frac{A-T}{T}\right|\times100}\end{aligned}\]

Largest true value

3025 55 55
\[\begin{aligned}T_1&=55^2=3025\\[6pt]\class{mjx-pe-s7}{\%\text{ error}}&\class{mjx-pe-s7}{{}=\left|\frac{2500-3025}{3025}\right|\times100}\\[6pt]&\class{mjx-pe-s7}{{}=\class{mjx-pe-key}{\boldsymbol{17.4\%}}}\end{aligned}\]

Maximum percentage error = 23.5%. The smallest true value gives the bigger error, because we divide by it.